Chop the tomatoes (large dice) and rough chop the basil; place all the ingredients into a blender, in batches, and process only a few seconds so that you get a coarse chop, not a pulp.
Taste and adjust seasonings.
Place all the ingredients into a cheesecloth bag that you fashion, and suspend over a bowl for 6 hours or overnight; discard the first few red drops, then the liquid will be clear. Do not squeeze the bag or you will color the liquid.
Spoon into individual serving bowls and garnish as desired.